AN IMPROVED PISO ALGORITHM FOR THE COMPUTATION OF …webx.ubi.pt/~pjpo/ri23.pdf · 2008. 5. 20. ·...

AN IMPROVED PISO ALGORITHM FOR THECOMPUTATION OF BUOYANCY-DRIVEN FLOWS

Paulo J. OliveiraDepartmento de Engenharia ElectromecaÃnica, Universidade da Beira Interior,

CovilhaÄ , Portugal

Raad I. IssaDepartment of Mechanical Engineering, Imperial College of Science,

Technology and Medicine, London, UK

A numerical procedure for the calculation of buoyancy-driven �ows using the �nite-volumeapproach is presented. It is based on an extension of the operator-splitting procedure PISOof Issa [1] to the speci�c case in which the coupling between velocity=pressure and tem-perature is important, as is the case in problems involving free-convection �ows. A com-parison of the proposed procedure with a standard iterative method shows improvementboth in terms of computing speed (a factor of 2.1 to 4.1) and robustness.

1. INTRODUCTION

Buoyancy-driven or natural-convection ¯ows are those generated by densitygradients, which in most cases arise from some imposed external heat source. These¯ows are typical in atmospheric science applications but occur in many engineeringapplications as well. The challenge is that most (not to say all) numerical methodsdesigned to handle the pressure±velocity coupling, based for example on the SIM-PLE procedure of Patankar and Spalding [2] or other segregated method, whenapplied to buoyant ¯ows result in a considerable increase in the number of iterationsfor convergence as the strength of buoyancy raises. This di�culty in the computationof free-convection ¯ows has been recognized for a long time (Caretto et al. [3]), but itis fair to acknowledge that the e�orts to overcome it have not been very successful.In fact, a survey of the literature reveals that what most authors having to deal withproblems involving buoyancy tend to do is just to append the energy equation to themomentum=pressure solution procedure, and hope that the sequential method willeventually converge (e.g., Jang et al. [4]). Important exceptions are the work ofGalpin and Raithby [5] and, more recently, that of Sheng et al. [6±7], which arediscussed below.

The main purpose of this work is the development of a better numericalmethod for the computation of buoyant ¯ows. A simple ¯ow geometry is chosen as

Received 8 March 2001; accepted 27 July 2001.Address correspondence to Dr. P. J. Oliveira, Universidade da Beira Interior, Departamento de

Enga ElectromecaÃnica, 6201-001 CovilhaÄ, Portugal. E-mail: [email protected]

Numerical Heat Transfer, Part B, 40: 473±493, 2001Copyright # 2001 Taylor & Francis1040-7790 /01 $12.00 + .00

473

test case: the two-dimensional square cavity with vertical walls heated at two dif-ferent temperatures and horizontal walls adiabatic (see Figure 1). This problem hasbeen extensively investigated because of the following reasons. (1) It models manyreal situations with interest to di�erent engineering ®elds: double glazing, glass

NOMENCLATURE

aP; am ; a0 coe�cients in the discretizedequations

cp speci®c heatC`; CL Courant number (local and global)g gravity accelerationH convective=di�usive operator

[H(¿) =P

am¿m ]k conductivityL side of square cavityNu Nusselt number (= hL =k)p pressureqw wall heat ¯uxPr Prandtl number (= mcp=k)Ra Rayleigh number

(= gb DT L 3=an)S source term in the discretized

equationst timeT temperatureui velocity componentsxi spatial coordinatesa thermal di�usivity

au underrelaxation factor formomentum

b coe�cient of thermal expansiondx control-volume sizedt time stepDT temperature di�erence (= Th ± Tc)Di gradient operatorm; n dynamic and kinematic viscosityr densityV volume of a control volume

Subscripts and superscripts

h; c; 0 hot, cold, and reference(temperatures)

i; j Cartesian componentsP; m main control volume and its

neighborsui ; T velocity, temperature (coe�cients;

H operator)n; n ‡ 1 previous and present time levels¤; ¤¤; ¤¤¤ intermediate values in algorithm

Figure 1. Sketch of the £ow geometry.

474 P. J. OLIVEIRA AND R. I. ISSA

melting furnaces, motion of magma within tectonic plates (geophysical and astro-physical interest), cooling of some types of nuclear reactors, solar panels, roomventilation by natural convection, etc. (2) . The apparent physical simplicity hides thedi�cult challenge to the numerical methods mentioned above: coupling of the energyand hydrodynamics equations becomes very sti� as the Rayleigh number (Ra) in-creases, resulting in severe numerical instability di�culties.

This article is motivated by the latter reason, and the idea is to extend theoperator-splitting algorithm PISO (Issa [1]) to the case of free-convection ¯ows, inwhich there is a three-variable coupling: velocity, pressure, and temperature. Beforeformulating and explaining the numerical procedure developed here, a short reviewof previous relevant numerical work is presented.

Most of the early simulations of natural convection in a rectangular cavitywere based on the vorticity=streamfunction formulation, and a classical reference onthat is the work of de Vahl Davis [8]; a more up-to-date review of that, and otherwork with the primitive-variables formulation, is provided in the article of Barakoset al. [9]. During the past 5 years or so, that problem has been subjected to studiesusing ®nite-di�erence methods with parallel computation [10], ®nite-element meth-ods (FEM) of the coupled Galerkin type [11], or decoupled type with time factor-ization [12] and pressure-correction techniques [13], and ®nite-volume methods(FVM) in nonstaggered meshes with various algorithms: SIMPLE [14, 15], SIM-PLER [16], and PISO with unstructured meshes [17]. In all these studies the energyequation is either solved in a coupled way with the other equations (in FEM) or it isappended to the iteration loop and solved sequentially at the end of the pressure±velocity algorithm. Emphasis has often been placed on obtaining very accurateresults which can be used for bench marking, e.g., Hortmann et al. [14] for Ra = 104±106, Le Quere [18] for Ra = 106±108, SyrjaÈ laÈ [11] for Ra = 104±107, and Nonino et al.[13] for Ra = 105±108. Of particular note is the work of Hortmann et al. [14], whoused very ®ne meshes (as ®ne as 6406640) together with the full multigrid method.The multigrid method is a numerical technique that can be applied to any ¯uid ¯owproblem and has nothing speci®c to free-convection ¯ows.

A close look to the above works reveals a common feature: buoyancy e�ectslead to considerable requirements in terms of CPU time and computer memory,especially at high Ra. For example, in [12] it is mentioned that 24 h of CPU time arerequired to solve the problem with a 166-MHz Pentium computer in a 16616 FEMmesh at the low value of Ra = 103 (a similar computation with the present methodtakes a few seconds). In [13], from 2,500 up to 9,000 iterations are required, for Rafrom 105 to 108, with time steps as small as 5610± 6 L 2=a (here some 400 iterationswith dt = 10± 4 are su�cient). In [11] it is stated that di�culty and cost increaseconsiderably when Ra goes from 106 to 107. It thus appears that a better way ofintegrating the energy equation into the ¯ow solver is required, so that the inter-linkage between the temperature and velocity ®elds is more e�ectively accounted forand hence results in a less steep increase in computer time as Ra increases.

There are not many studies with emphasis on improving the numerical schemefor buoyant ¯ows. The numerical treatment of the temperature±velocity coupling isaddressed by Galpin and Raithby [5], who devised a better Newton-like linearizationof the advective ¯uxes in the energy equation which led to a tighter treatment of thecoupling between the governing equations. They then used a special coupled solution

AN IMPROVED PISO ALGORITHM 475

method to solve for u; p; and T at a given node, without relying on the SIMPLEpressure-correction algorithm. Galpin and Raithby’s method compared favorablywith others, showing improved convergence rate for ®ne meshes, but its blocklikenature makes it less attractive for general applications and di�cult to implement intoexisting segregated solvers. In addition to that work, an attempt to improve theSIMPLE algorithm for buoyant ¯ows was recently given by Sheng et al. [6] and [7].These authors relate the velocity corrections to both pressure and temperature (ordensity) variations, but while a pressure-correction equation can be derived in theusual way, the temperature variation results from an additional solution of the en-ergy equation, after the momentum prediction. The results presented in [7], and in [6]for high Ra, show hardly any improvement over the simpler sequential solution ofthe energy equation at the end of the iterative cycle. A possible reason for this de-®cient behavior may be related to the fact that the intermediate energy equation issolved with a velocity ®eld which does not satisfy continuity.

As mentioned above, the base method in the present study is the PISO al-gorithm of Issa [1]. If the energy equation is merely appended to the base algorithm,then there is no improvement in the calculation of free-convection ¯ows, as foundby Jang et al. [4], who compared the performance of several algorithms, SIMPLER[19], SIMPLEC [20], and PISO. However, the operator-splitting nature of PISO,with various velocity=pressure correction stages following the momentum predic-tion, makes it much more amenable to di�erent arrangements for the solution ofthe energy equation, with the possibility of temperature corrections. A carefulchoice of the algorithm results in better numerical behavior, as the present resultswill show.

2. FORMULATION OF THE PROBLEM

The objective is to develop an e�cient numerical method for the solution of themotion and energy equations of an incompressible Newtonian ¯uid whose move-ment is caused by natural convection. The ¯ow is assumed to be laminar and theBoussinesq approximation valid, so that the continuity, momentum, and energyequations become

q

qxi(rui) = 0 (1)

q

qt(rui) ‡ q

qxj(rujui) = ±

qpqxi

‡ q

qxjm

qui

qxj± rb(T ± T 0)gi (2)

q

qt(rcpT ) ‡ q

qxj(rcpujT ) =

q

qxjk

qTqxj

(3)

In these equations the dependent variables are the velocity components, ui, pressure,p, and temperature, T . The physical properties of the ¯uid are the density, r, visc-osity, m, speci®c heat at constant pressure, cp, thermal conductivity, k, and the vo-lumetric factor of thermal expansion, b, which are all assumed constant. The ¯owoccurs inside the square cavity represented in Figure 1 and the only nonzero com-ponent of gravity is g2 ² ± g. Although the interest is only on steady-state solutions,


the time-dependent terms are retained in Eqs. (2) and (3) because the numericalscheme is based on a time-marching advancement procedure.

Boundary conditions for Eqs. (1)±(3) are as follows: no slip applies on the foursurrounding walls; imposed cold and hot temperatures on the vertical walls, T = T c

(² T 0), at x = 0, and T = T h at x = L ; and a zero heat ¯ux on the horizontal walls,qT=qy = 0 at y = 0 and y = L .

This problem is characterized by three nondimensional numbers: Prandtl,Pr = mcp=k; Rayleigh, Ra = gb DT L 3=(na); and Nusselt, Nu = hL =k = qwL =(DTk),where DT ² (T h ± Tc), qw is the heat ¯ux across the vertical walls, and h is thecoe�cient of heat transfer. The strength of the coupling between the momentum andenergy equations is quanti®ed by the Rayleigh number, as shown by the non-dimensional form of the governing equations. When Pr ¶ 1 the usual practice is tochoose a=L as the characteristic velocity, so that the steady form of the non-dimensional y-momentum equation becomes

q

qX(UV ) ‡ q

qY(V V ) = ±

qPqY

‡ Prq2VqX2

‡ q2VqY 2

‡ Ra ¢ Pr ¢ y

It is evident from this equation that for constant Pr (here we use Pr = 1 or 0.71), thefree-convection source term ( = Ra Pr y, y = (T ± T 0)=DT ) becomes dominant asthe Rayleigh number Ra is increased, leading to a set of coupled sti� partial dif-ferential equations. This term should therefore be treated as implicitly as possible toavoid numerical instabilities; this is the objective of the present work, and to this aima new arrangement of the operator-splitting procedure PISO [1] has been devised,as explained in the next section.

3. NUMERICAL PROCEDURE

The set of equations (1)±(3) is transformed into a set of algebraic equationsafter application of any standard discretization procedure in a staggered mesh (e.g.,Patankar [19]). As mentioned in [1], the numerical algorithm is basically independentof the particular di�erencing scheme chosen; here the backward, fully implicitscheme is used to represent the time derivatives, and the hybrid and central di�er-encing schemes are used to approximate the convective and di�usive spatialderivatives, respectively. With the operator notation introduced by Issa [1], the ®nite-di�erence counterpart of the momentum (2) and energy (3) equations is

rVdt

‡ aui0 uiP = H(ui) ± Dip ‡ Sui ‡ rV

dtun

iP (4)

rVdt

‡ aT0 TP = HT (T ) ‡ ST ‡ rV

dtT n

P (5)

In the above equations, subscript P denotes values at the central node of the com-putational molecule and H stands for the operator which accounts for the in¯uenceof neighbor nodes, H(ui) =

Paui

mum (m from 1 to number of neighboring cells, 4 in2-D). The coe�cients am are composed by convective and di�usive ¯uxes across cellfaces, its sum is given by a0 =

Pam , and Di is the ®nite-di�erence representation of


the gradient operator. The source terms S include all the remaining terms, which formomentum comprise the free-convection contribution, given by rgbV(TTP ± T 0),where TT P is the temperature interpolated at velocity point P. The computational timestep is denoted by dt, previous time-level values are denoted with n, and V representsthe volume of a computational cell.

The continuity equation (1) is used to obtain an equation for pressure bytaking the divergence of the discretized momentum equation.

In the remainder of this section, we give ®rst the best algorithm to emerge fromthe numerical experiments (to be discussed in the Results section) and then we brie¯ydescribe other variants that have been tested. The in¯uence of the solver on thealgorithm performance is discussed at the end of the section.

3.1. Proposed Solut ion Algorithm

A form of the operator-splitting PISO algorithm of Issa [1] is now presentedwhich embeds the temperature variation of the momentum source Sui into the ve-locity±pressure coupling treatment. Following the original presentation, the di�erentpredictor-corrector level of variables and coe�cients are denoted with superscripts n,*, **, ***. The central coe�cient is denoted aP = a0 ‡ rV=dt.

Step 1. Momentum predictor.

(auiP )nu¤

iP = Hn(u¤i ) ± Dipn ‡ (Sui )n ‡ rV

dtun

iP (6)

This implicit equation is to be solved for the velocity components at all controlvolumes, u¤

i .

Step 2. First velocity corrector.

(auiP )nu¤¤

iP = Hn(u¤i ) ± Dip¤ ‡ (Sui )n ‡ rV

dtun

iP (7)

Note that velocity in the H operator and in the term on the left-hand side areevaluated at di�erent iteration levels, hence the terminology `̀ operator splitting. ’’

Step 3. First pressure-correct ion equat ion. After subtraction of Eq. (6)from Eq. (7), division by aui

P , and making use of the discretized form of the continuityequation

Di(ru¤¤i ) = 0 (8)

the ®rst pressure-correction equation is obtained:

Dir

(auiP )n Di( p¤ ± pn) = Di(ru¤

i ) (9)

These ®rst three steps are part of the standard PISO and also correspond to theSIMPLE algorithm.


Step 4. Temperature predictor.

(aTP)¤T ¤

P = HT ¤(T ¤) ‡ (ST )n ‡ rVdt

T nP (10)

where HT¤(T ) is based on coe�cients (aTm)¤, which are to be calculated with the

corrected velocities u¤¤i , and the equation indicates an implicit solution for the

temperature ®eld, thus requiring a linear equation solver.

Step 5. Second velocity corrector. Step 5 is based on the following opera-tor-splitting equation:

(auiP )nu¤¤¤

iP = Hn(u¤¤i ) ± Dip¤¤ ‡ (Sui )¤ ‡ rV

dtun

iP (11)

where it is important to note that the free-convection source term is based on thenewly computed temperature T ¤, from Eq. (10). Steps 4 and 5 depart from thestandard PISO.

Step 6. Second pressure-cor rect ion equat ion. Following a procedure si-milar to the described above, the second pressure-correction equation is obtained:

Dir

(auiP )n Di( p¤¤ ± p¤) = Di

r

(auiP )n [(S ui¤ ± S uin) ‡ H n(u¤¤

i ± u¤i )] (12)

where the triple-starred velocities are forced to satisfy the continuity equation:

Di(ru¤¤¤i ) = 0 (13)

The important point here is that the pressure-correction equation (12) now includes aterm which arises from the variation of the free-convection source term with thetemperature. Hence a tighter coupling between the velocity and temperature ®eldsis achieved.

Step 7. First temperature corrector.

(aTP)¤¤T ¤¤

P = HT¤¤(T ¤) ‡ (ST )n ‡ rVdt

T nP (14)

where HT¤¤(T ) is based on the coe�cients (aTm)¤¤, which are calculated with the

corrected velocities u¤¤¤i . This equation is solved pointwise and does not require a call

to the solver.Steps 1±7 de®ne the proposed algorithm, which is an optimized version of the

operator-splitting procedure for free-convection ¯ows. These steps are successivelyapplied at each time level and the ®elds given by p¤¤, u¤¤¤

i , and T ¤¤ correspond, towithin a certain truncation error [1], to those of the next time step, p n‡1, u n‡1

i , andT n‡1. The steady-state solution is thus approached by a sequential time-marchingprocedure which does not require iteration within each time step. Proximity to thesteady-state solution is assessed by controlling the maximum residuals of all equa-tions; these residuals are computed from newly obtained coe�cients and previoustime-level variables.


3.2. Other Algorithms

Numerical experiments revealed that the algorithm described in Section 3.1yields the best results, in the sense of less computational time and wider range ofstability. Its performance will be compared with a standard procedure in Section 4.It is, however, worthwhile to mention brie¯y other algorithm variants which havebeen tested, and which are modi®cations or rearrangements of the above:

Variant 1. The temperature predictor is the first step, before the momentumpredictor, and the first temperature corrector comes after the first velocity corrector;it may, or may not, have a second temperature corrector at the end of the cycle. Thesequence of computed variables is

T ¤

u¤, v¤

p¤, u¤¤, v¤¤

T ¤¤

p¤¤, u¤¤¤, v¤¤¤

with, or without, T ¤¤¤

Variant 2. The temperature predictor occurs after the velocity predictor, andthen follows variant 1 (note that variation of the free-convection source Sui is in-cluded into the first corrector of ui):

u¤, v¤

T ¤

p¤; u¤¤; v¤¤ [including e�ect of (Su)¤ ± (Su)n]T ¤¤

p¤¤; u¤¤¤; v¤¤¤

with, or without, T ¤¤¤

Variant 3. This variant basically follows the proposed algorithm except thetemperature corrector (step 7), which is solved implicitly, therefore being more ap-propriately called a second temperature predictor.

Variant 4. Standard PISO (as in [1]) with the temperature equation solved se-quentially at the end of the cycle.

Variant 5. All the above arrangements (variants 1±4) have been tested with re-calculation of the velocity coefficients; i.e., (aui

P )n and Hn(u¤¤i ) in Eq. (11) are replaced

by (auiP )¤ and H¤(u¤¤

i ).

An assessment of these algorithm variants, based on numerical experiments,may be summarized as follows:

° Variant 2 did not perform well, a plausible cause being the fact that thetemperature prediction is based on a velocity ®eld which does not satisfycontinuity.

° Variant 1 performed better than variant 4, but not as well as the proposedalgorithm. In general, variant 4 (the standard PISO) did not o�er signi®cant


advantages over SIMPLEC, when both algorithms were used with iterativemarching (with underrelaxation factors) , thus con®rming the ®ndings of Janget al. [4]. With time marching, the standard PISO would tend to work betterthan SIMPLEC but, as mentioned above, not up to the proposed algorithm.

° Variant 3 shares some of the good behavior of the proposed algorithm buthas a drawback: the point of minimum CPU time presents a stronger de-pendence on the grid size and Courant number.

° Recalculation of the coe�cients aui in variant 5 was always penalized by thetime necessary to perform that operation (5 times more than 1 iteration ofthe conjugate-gradient solver).

3.3. Solut ion of the Linear Sets of Equat ions

Several sets of linear algebraic equations must be solved at each time step, andthe overall performance of the numerical algorithm depends on the method used tosolve them. It is well known that the pressure-correction equations should be solvedto a tight tolerance if propagation of errors resulting from a velocity ®eld failingto satisfy the divergence-free condition is to be avoided. This requirement, togetherwith the Poisson-like behavior and Neumann-type boundary conditions of thepressure-correction equation, usually result in a great number of inner iterations ateach time step, so an e�cient solver is required. The conjugate gradient solver forsymmetric matrices (CGS) o�ers a good compromise between simplicity and e�-ciency. The convergence rate of this method is high, especially when linked to anappropriate preconditioner. In the present study the version of CGS with in-complete-Choleski preconditioner given by [21] was adopted for both the PISO andSIMPLEC. The simpler application of line-by-line iteration with the tridiagonalmatrix algorithm was used for the other equation sets, which are much easier to solvesince the upwind di�erencing scheme leads to coe�cient matrices in which some ofthe elements are zero when convection dominates. These matrices are also no longersymmetric, as is the case of the matrix for the pressure equation, and thus conjugate-gradient solvers become more involved.

The number of inner iterations required to solve each of the linear equationsets may be ®xed, as recommended by Patankar [19] with the argument that theequations need not be solved very precisely since the overall procedure is itselfiterative, or may vary according to a speci®ed tolerance for the relative decay of theresiduals (e.g., [20]). Numerical experiments showed that the latter approach is moree�cient and so it is adopted here. The number of inner iterations necessary to satisfya prede®ned residual tolerance denoted by g depends on the Rayleigh number andtends to increase for low Rayleigh numbers (Ra= 103 and 104), since the importanceof di�usion is then accentuated. It is also interesting to mention that the number ofinner iterations to solve the energy equation was greater than for the momentumequations. The reason for this is related to the di�erent type of boundary conditionsfor each variable: Neumann type for T and Dirichlet for ui. The value g used to stopthe inner iterations (when residual=initial residual < g) was optimized so as to mini-mize the computing time; the value g = 0.25 was found to o�er the best results for themomentum and energy equations, and values of g = 0.1 and 0.05 were found for thepressure-correction equations, respectively, for Ra > 104 and µ 104.


4. RESULTS

Assessment of the proposed method has been carried out by performing a greatnumber of computer runs, where relevant parameters were individually varied, andfrom analysis of the resulting CPU times in terms of a nondimensional time step. Themethod chosen for comparison is the iterative SIMPLEC algorithm, implemented asdescribed in its original reference, Van Doormaal and Raithby [20]. All the ar-rangements of the operator-splitting procedure described in Section 3, and also thecomparison method, have been implemented into a single computer code, based onthe staggered-mesh arrangement, in such a way that one could easily switch methodsby proper choice of a parameter. Hence it is assured that CPU times are not falsi®edby di�erent programming. A PC Pentium Pro computer with a processor running at200 MHz was used in all computations.

Runs were made for Ra ranging from 103 to 108; higher Ra were not testedsince physical instabilities and transition to turbulent ¯ow are then expected to occurand no turbulence model has been used. Furthermore, we are interested only indetermining the numerical behavior of the algorithm to reach a steady-state solution.The results to be presented ®rst correspond to a Prandtl number of unity (Pr = 1) anda square cavity. A second case was considered with Pr = 0.71 (valid for air), whichallows for a direct comparison with results from other authors. A grid with 22622nodes was initially adopted, but it was soon realized that proper assessment of themethod for high Rayleigh numbers could not be done in such grid since it cannotresolve the ¯ow pattern. This can be inferred from observation of a global parametercharacterizing the resulting ¯ow ®eld obtained with di�erent grids. Figure 2 showsthe predicted Nusselt number as a function of the number of internal cells in the grid(N); it is seen that only the ®ne grids, 30630 (N= 784) and 40640 (N= 1,444) , areable to yield a Nusselt number independent of N and with su�cient accuracy. Thecomputational performance of di�erent numerical methods cannot be comparedadequately if the ¯ow ®eld is not well resolved, otherwise the relative performancemay change when the grid is re®ned. Higher accuracy can be achieved with non-uniform grid spacing, as in the second case discussed below.

The time-marching behavior of the modi®ed PISO method compared with theiterative SIMPLEC can be seen in Figures 3±6, which correspond to Rayleighnumbers from 104 to 107. Curves of processor time (CPU) against global Courantnumber are given for two grids, 30630 and 40640. The global Courant number(CL ) is de®ned with a typical velocity (U, as in [5]) and the cavity width, that is,

CL =dt U

Lwith U ² a

LRa

1 ‡ 1=Pr

1=2

(15)

This de®nition for characteristic velocity is very convenient, since U can be readilydetermined for a given Ra and Pr; a more precise, but problem-dependent , char-acteristic velocity could be de®ned with the maximum velocity in the resulting so-lution ®eld. However, it was found that its value correlates well with the abovechoice for U, being umax º U=3. Since the grids used in this ®rst test case wereuniform and square, a local Courant number can be obtained from C` =

dt ¢ U=dx = Ni ¢ CL , where Ni is the number of internal cells along x or along y.


For the iterative method the curves were obtained by varying the underrelaxationfactor in the momentum equation, au, and determining an equivalent time step fromthe analogy between time-marching and iterative approaches (see [1] and [20]):

dt =au

1 ± au

rVau

0(16)

This is sometimes called a ``pseudo’’ or ``distorted ’’ time step because while for atime-marching algorithm dt is constant, the equivalent time step in an iterativemarching algorithm varies from point to point, as shown by Eq. (16). In order to

Figure 2. E¡ect of mesh size on predicted Nusselt number.


obtain a single value for the equivalent time step for the iterative algorithm, the lastterm in Eq. (16) is evaluated as Min(rV=au

0) .Figures 3±6 show that the new algorithm is not only faster than SIMPLEC but

also presents, in general, a wider range of stability. This range becomes narrowerwhen the Rayleigh number increases, as would be expected. Inspection of Figures 3±6 also shows that the Courant number corresponding to the point of minimumcomputing time is approximately independent of both the Rayleigh number and thegrid size. This optimum Courant number is roughly equal to CL º 1:8. As a con-sequence, one is able to start a computation, without any preliminary trials, using analmost optimum time step given by

Figure 3. Comparison of time-marching behavior for Ra = 104.


(dt)opt º 1:8 NidxU

(17)

In terms of a local Courant number de®ned with the maximum local velocity[C`max = dt ¢ Max(u=dx)], which is a more appropriate measure of the stabilitybrought about by the present implicit algorithm, the points of minimum CPU timein Figures 3±6 have values ranging from 45.4 at Ra = 104 to 25.3 at Ra = 107.

Figures 7a and 7b show the minimum CPU times achieved with optimizedvalues of dt and au using the modi®ed PISO method and SIMPLEC, as a function ofthe Rayleigh number for two grids. It is seen that for the ®ner grid, gains in CPU



time are quite substantial. In relative terms, the new method is faster than SIMPLECby a factor ranging from 1.5 to 2.8, when both methods are optimized. In actualcomputations of free-convection ¯ows this factor will be higher, since it is notpractical to optimize the methods before the actual runs, and SIMPLEC shows avery sharp dependence of the optimum point on the underrelaxation factor au

(Figures 3±6). With SIMPLEC, any departure of au from the optimum value resultsin a considerable increase of CPU time from its minimum level. The increase in CPUtime seen in Figure 7a when Ra drops from 104 to 103 can be explained as due tothe onset of di�usion e�ects. At low Ra, di�usion e�ects are dominant over con-vection and the energy and momentum equations become like Poisson equations



(di�usion-dominated equations), with the consequence that many inner iterationswithin the solver are required to bring the residuals below the prespeci®ed tolerance.

A second case with Pr = 0.71 was simulated on a nonuniform grid of 42642nodes (40640 internal control volumes) designed so that relatively accurate pre-dictions could be achieved. Grid spacing is smallest near the walls, with a minimumvalue of dxmin = 0:00488L , and expands from there toward the cavity center at aconstant geometric rate of 15%. Although the purpose of the present study is notrelated with ``accuracy ’’Ðotherwise a better di�erencing scheme would have beenused to represent the convective=advective terms in the equationsÐthere is someinterest in documenting the resulting Nusselt numbers for this case: Nu = 1.117,



2.244, 4.513, 8.755, 16.423, and 30.473 at, respectively, Ra = 103, 104, . . . , 108. Theseresults compare very favorably with the various benchmark data available in theliterature (consistent values in [14, 18, 11, and 13]), with relative di�erences rangingfrom 0.04% (Ra = 104), 0.2% (Ra = 105), to 0.8% (Ra = 108). We may thus concludethat the present grid is adequate to provide results up to Ra = 108 with su�cient``engineering’’ accuracy (below 1%) and we turn to the main question of algorithme�ciency. Figure 8 shows the minimum CPU times for those runs as a function of

Figure 7a. Minimum processor time as a function of Ra: grid 40640.


Ra, for the new PISO algorithm and the standard SIMPLEC. The rate of increase incomputer time with the Rayleigh number is much steeper for SIMPLEC (comparewith Figure 7a for the uniform grid), and the new PISO is faster than SIMPLEC by afactor ranging from 2.9 at Ra = 105 to 5.1 at Ra = 108. The maximum local Courantnumber for these optimum runs varied from Clmax = 19:2 at Ra = 105 to 14.4 atRa = 108, showing an expected tendency to decrease as Ra is raised. However, thatvalue for the optimum Courant number is always over 10, re¯ecting the good sta-bility of the method, and the range of variation is not too wide.

Figure 7b. Minimum processor time as a function of Ra: grid 30630.


Figure 9 shows the variation of CPU time with the global Courant number forthese runs on the nonuniform grid in which good accuracy is achieved. The optimumtime step decreases with Ra, but a time step such that CL º 0:9 yields almost opti-mum computing times for all cases. Instead of taking a ®xed optimum CL , as sug-gested above for the uniform-grid cases, a more re®ned ®tting is given by thecorrelation

CL ;opt = 14:8 Ra± 0:153 (18)

Figure 8. Minimum processor time as a function of Ra for the case Pr = 0.71 (grid 42642 nonuniform).


which can be used to estimate the optimum time steps; this function is represented bythe dashed lines in Figure 9.

5. CONCLUSIONS

A numerical algorithm especially designed to solve the equations for buoyancy-driven ¯ows has been presented. It is based on the idea of operator splitting, whichwas originally introduced to deal with the pressure±velocity coupling present in the¯uid motion equations, and it is here extended to cater for the additional strongcoupling in buoyant ¯ows, between temperature and velocity. The implicit treatment

Figure 9. T|me-marching behavior of new PISO for the case Pr = 0.71, nonuniform grid 42642. Dashedline correlates points of minimum CPU time.


of the coupling term, essential to dump the numerical instabilities, has been achievedby embedding the calculation of temperature into the predictor-corrector steps ofPISO, as follows:

° The temperature equation is solved implicitly after the ®rst velocity-correc-tor step of PISO.

° The new temperature ®eld changes the buoyancy term in the momentumequations; this change is incorporated into the second velocity corrector andthe pressure equation of PISO.

° A ®nal temperature correction is added to the usual PISO so that tem-perature is adjusted to the new velocity ®eld.

The modi®ed PISO method is assessed by comparison with a standard iterativeprocedure (SIMPLEC), using the test problem of buoyant ¯ow in a square cavitywith di�erentially heated vertical walls, for several Rayleigh numbers (up to 108) andgrid sizes (up to 42642 nonuniform). The results show the new method to be betterin terms of both computing speed and stability. There is an improvement in CPUtime by a factor ranging from 2.1 (uniform grid) to 4.1 (nonuniform grid, morerealistic simulations), on average, when both methods are optimized. This speed-upfactor tends to increase sharply when conditions depart from optimal, a sign ofimproved stability. Indeed, the range of time steps for which the new free-convectionPISO enables converged results is much wider than the corresponding equivalenttime-step range for SIMPLEC, a measure of improved stability.

Practical usage of the proposed method is facilitated by guidance on the valuesto assign to the time step in order to have minimum computing times.

REFERENCES

1. R. I. Issa, Solution of the Implicitly Discretized Fluid Flow Equation by OperatorSplitting, J. Comput. Phys., vol. 62, pp. 40±65, 1986.

2. S. V. Patankar and D. B. Spalding, A Calculation Procedure for Heat, Mass, and Mo-mentum Transfer in Three-Dimensional Parabolic Flows, Int. J. Heat Mass Transfer,vol. 15, pp. 1787±1806, 1972.

3. L. S. Caretto, A. D. Gosman, and D. B. Spalding, Removal of an Instability in a FreeConvection Problem, Rep. EF=TN=A=35, Imperial College, London, 1971.

4. D. S. Jang, R. Jetli, and S. Acharya, Comparison of the PISO, SIMPLER, and SIMPLECAlgorithms for the Treatment of the Pressure-Velocity Coupling in Steady Flow Pro-blems, Numer. Heat Transfer, vol. 10, pp. 209±228, 1986.

5. P. F. Galpin and G. D. Raithby, Numerical Solution of Problems in Incompressible FluidFlow: Treatment of the Temperature-Velocity Coupling, Numer. Heat Transfer, vol. 10,pp. 105±129, 1986.

6. Y. Sheng, M. Shoukri, G. Sheng, and P. Wood, A Modi®cation of the SIMPLE Methodfor Buoyancy-Driven Flows, Numer. Heat Transfer B, vol. 33, pp. 65±78, 1998.

7. Y. Sheng, M. Shoukri, G. Sheng, and P. Wood, New Version of SIMPLET and ItsApplication to Turbulent Buoyancy-Driven Flows, Numer. Heat Transfer A, vol. 34,pp. 821±846, 1998.

8. G. de Vahl Davis, Natural Convection of Air in a Square Cavity: A Bench Mark Nu-merical Solution, Int. J. Numer. Meth. Fluids, vol. 3, pp. 249±264, 1983.


9. G. Barakos, E. Mitsoulis, and D. Assimacopoulos, Natural Convection Flow in a SquareCavity Revisited: Laminar and Turbulent Models with Wall Functions, Int. J. Numer.Meth. Fluids, vol. 18, pp. 695±719, 1994.

10. P. Wang and R. D. Ferraro, Parallel Computation for Natural Convection, Concurrency:Practice and Experience, vol. 9, pp. 975±987, 1997.

11. S. SyrjaÈ laÈ, Higher Order Penalty-Galerkin Finite Element Approach to Laminar NaturalConvection in a Square Cavity, Numer. Heat Transfer A, vol. 29, pp. 197±210, 1996.

12. B. BermuÂdez and A. NicolaÂs, An Operator Splitting Numerical Scheme for Thermal=Isothermal Incompressible Viscous Flows, Int. J. Numer. Meth. Fluids, vol. 29, pp. 397±410, 1999.

13. C. Nonino and G. Comini, An Equal-Order Velocity-Pressure Algorithm for In-compressible Thermal Flows, Part 1: Formulation; and Part 2: Validation, Numer. HeatTransfer B, vol. 32, pp. 1±15 and pp. 17±35, 1997.

14. M. Hortmann, M. PericÂ, and G. Scheuerer, Finite Volume Multigrid Prediction of La-minar Natural Convection: Bench-mark Solutions, Int. J. Numer. Meth. Fluids, vol. 11,pp. 189±207, 1990.

15. M. Char and Y. Hsu, Computation of Buoyancy-Driven Flow in an Eccentric CentrifugalAnnulus with a Non-orthogonal Collocated Finite Volume Algorithm, Int. J. Numer.Meth. Fluids, vol. 26, pp. 323±343, 1998.

16. G. Croce, G. Comini, and W. Shyy, Incompressible Flow and Heat Transfer Computa-tions Using a Continuous Pressure Equation and Nonstaggered Grids, Numer. HeatTransfer B, vol. 38, pp. 291±301, 2000.

17. Y. G. Lai, An Unstructured Grid Method for a Pressure-Based Flow and Heat TransferSolver, Numer. Heat Transfer B, vol. 32, pp. 267±281, 1997.

18. P. Le Quere, Accurate Solution to the Square Thermally Driven Cavity at High RayleighNumber, Comput. Fluids, vol. 20, pp. 29±41, 1991.

19. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.20. J. P. Van Doormaal and G. D. Raithby, Enhancement of the SIMPLE Method for

Predicting Incompressible Fluid Flows, Numer. Heat Transfer, vol. 7, pp. 147±163, 1984.21. J. A. Meijerink and H. A. Van Der Vorst, Guidelines for the Usage of Incomplete

Decompositions in Solving Sets of Linear Equations as They Occur in Practical Problems,J. Comput. Phys., vol. 44, pp. 134±155, 1981.


AN IMPROVED PISO ALGORITHM FOR THE COMPUTATION OF …webx.ubi.pt/~pjpo/ri23.pdf · 2008. 5. 20. ·...

Documents

Transcript of AN IMPROVED PISO ALGORITHM FOR THE COMPUTATION OF …webx.ubi.pt/~pjpo/ri23.pdf · 2008. 5. 20. ·...